# Adjoint-based SQP Method with Block-wise quasi-Newton Jacobian Updates for Nonlinear Optimal Control

## TL;DR

This paper introduces an adjoint-based SQP method with block-wise quasi-Newton Jacobian updates for nonlinear MPC, offering improved convergence and computational efficiency in solving complex non-convex optimization problems.

## Contribution

It presents a novel local convergence analysis and an efficient implementation of an adjoint-based SQP algorithm using block-structured TR1 quasi-Newton updates for nonlinear optimal control.

## Key findings

- Convergence analysis confirms local convergence properties.
- The proposed method reduces computational load in implicit discretizations.
- Simulation case studies demonstrate improved performance in nonlinear MPC.

## Abstract

Nonlinear model predictive control~(NMPC) generally requires the solution of a non-convex optimization problem at each sampling instant under strict timing constraints, based on a set of differential equations that can often be stiff and/or that may include implicit algebraic equations. This paper provides a local convergence analysis for the recently proposed adjoint-based sequential quadratic programming~(SQP) algorithm that is based on a block-structured variant of the two-sided rank-one~(TR1) quasi-Newton update formula to efficiently compute Jacobian matrix approximations in a sparsity preserving fashion. A particularly efficient algorithm implementation is proposed in case an implicit integration scheme is used for discretization of the optimal control problem, in which matrix factorization and matrix-matrix operations can be avoided entirely. The convergence analysis results as well as the computational performance of the proposed optimization algorithm are illustrated for two simulation case studies of nonlinear MPC.